The Method of Archimedes in the geometry of quadrics

Abstract

Confocal quadrics capture (encode) and geometrize spectral properties of symmetric operators. Certain metric-projective properties of confocal quadrics (most of them established in the first half of the XIX$^{\mathrm{th}}$ century) {\it carry out} (stick and transfer) by rolling to and influence surfaces {\it applicable} (isometric) to quadrics and surfaces geometrically linked to these, thus providing a wealth of integrable systems and projective transformations of their solutions. We shall mainly follow Bianchi's discussion of deformations (through bending) of quadrics. Interestingly enough, {\it The Method} of Archimedes (lost for 7 centuries and rediscovered in the same year as Bianchi's discovery (1906), so unknown to Bianchi) applies {\it word by word in both spirit and the letter} and may provide the key to generalizations in other settings.